Upper bounds on the multiplicative complexity of symmetric Boolean functions

Brandão, Luís T. A. N.; Çalık, Çağdaş; Turan, Meltem Sönmez; Peralta, René

doi:10.1007/s12095-019-00377-3

Cited by 5 publications

(9 citation statements)

References 15 publications

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“…A special metric that motivates Boolean functions is multiplicative complexity (MC): the minimum number of AND gateways, which is sufficient to implement a Boolean function based on {XOR, AND, NOT}. Paper [18] examines MC of the symmetrical Boolean functions, the output of which is invariant in the reordering of input variables. Based on the Hemming weights method, new methods are introduced that allow the synthesis of circuits with fewer logical elements of AND, compared to the upper limit.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Based on the Hemming weights method, new methods are introduced that allow the synthesis of circuits with fewer logical elements of AND, compared to the upper limit. Work [18] presents the generation of schemes for all such functions up to 25 variables. As a special focus, the authors report specific upper limits for MC of the elementary symmetric functions k n ∑ and counting functions k n ∑ up to n = 25 input variables.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…The reviewed literary sources [15][16][17][18][19][20][21][22][23][24] mainly report methods of minimizing symmetrical Boolean functions in the Boolean and Reed-Muller bases. There are methods for simplifying symmetrical Boolean functions that use theoretical objects of contiguous theory as Hemming scales, ordered binary decision-making diagrams (ROBDDs), etc.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Thus, the algorithms and methods, the software tools designed for them, covering the general procedure for minimizing symmetrical Boolean functions [15][16][17][18][19][20][21][22][23][24] and the method of figurative transformations take different approaches (principles of minimization). Therefore, they imply different prospects regarding the possibility of algorithmic minimization of symmetrical Boolean functions.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Thus, the classical analytical method still has the prospect of increasing its hardware capabilities in relation to the minimization of symmetrical Boolean functions. And this is the reason to believe that the software and technological base, which is represented by algorithms and methods with theoretical objects of adjacent theories [15][16][17][18][19][20][21][22][23][24], is insufficient to conduct theoretical research on the optimal minimization of symmetrical Boolean functions, in particular in the Reed-Muller basis.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

Solomko¹,

Tadeyev²,

Zubyk

et al. 2021

EEJET

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This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned function. This is a sufficient resource to minimize symmetrical Boolean functions that makes it possible to do without auxiliary objects, such as Karnaugh maps, cubes, etc. The perfect normal form of symmetrical functions can be represented by binary matrices that would represent the terms of symmetrical Boolean functions and the OR or XOR operation for them. The experimental study has confirmed that the method of figurative transformations that employs the 2-(n, b)-design, and 2-(n, x/b)-design combinatorial systems improves the efficiency of minimizing symmetrical Boolean functions. Compared to analogs, this makes it possible to enhance the productivity of minimizing symmetrical Boolean functions by 100‒200 %. There are grounds to assert the possibility of improving the effectiveness of minimizing symmetrical Boolean functions in the main and polynomial bases by the method of figurative transformations. This is ensured, in particular, by using the developed equivalent transformations of polynomial function conjuncterms based on the method of inserting similar conjuncterms followed by the operation of super-gluing the variables.

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

See 3 more Smart Citations

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

Solomko¹,

Tadeyev²,

Zubyk

et al. 2021

EEJET

View full text Add to dashboard Cite

show abstract

Boolean functions with multiplicative complexity 3 and 4

Çalık¹,

Turan²,

Peralta³

2020

Cryptogr. Commun.

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Garbled Circuits Reimagined: Logic Synthesis Unleashes Efficient Secure Computation

Yu,

Marakkalage,

De Micheli

2023

Cryptography

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Garbled circuit (GC) is one of the few promising protocols to realize general-purpose secure computation. The target computation is represented by a Boolean circuit that is subsequently transformed into a network of encrypted tables for execution. The need for distributing GCs among parties, however, requires excessive data communication, called garbling cost, which bottlenecks system performance. Due to the zero garbling cost of XOR operations, existing works reduce garbling cost by representing the target computation as the XOR-AND graph (XAG) with minimal structural multiplicative complexity (MC). Starting with a thorough study of the cipher-text efficiency of different types of logic primitives, for the first time, we propose XOR-OneHot graph (X1G) as a suitable logic representation for the generation of low-cost GCs. Our contribution includes (a) an exact algorithm to synthesize garbling-cost-optimal X1G implementations for small-scale functions and (b) a set of logic optimization algorithms customized for X1Gs, which together form a robust optimization flow that delivers high-quality X1Gs for practical functions. The effectiveness of the proposals is evidenced by comprehensive evaluations: compared with the state of the art, 7.34%, 26.14%, 13.51%, and 4.34% reductions in garbling costs are achieved on average for the involved benchmark suites, respectively, with reasonable runtime overheads.

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Upper bounds on the multiplicative complexity of symmetric Boolean functions

Cited by 5 publications

References 15 publications

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

Boolean functions with multiplicative complexity 3 and 4

Garbled Circuits Reimagined: Logic Synthesis Unleashes Efficient Secure Computation

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